Advertisement

Foundations of Physics

, Volume 40, Issue 9–10, pp 1419–1428 | Cite as

On the Complementarity of the Quadrature Observables

  • Pekka Lahti
  • Juha-Pekka Pellonpää
Article

Abstract

In this paper we investigate the coupling properties of pairs of quadrature observables, showing that, apart from the Weyl relation, they share the same coupling properties as the position-momentum pair. In particular, they are complementary. We determine the marginal observables of a covariant phase space observable with respect to an arbitrary rotated reference frame, and observe that these marginal observables are unsharp quadrature observables. The related distributions constitute the Radon transform of a phase space distribution of the covariant phase space observable. Since the quadrature distributions are the Radon transform of the Wigner function of a state, we also exhibit the relation between the quadrature observables and the tomography observable, and show how to construct the phase space observable from the quadrature observables. Finally, we give a method to measure together with a single measurement scheme any complementary pair of quadrature observables.

Complementarity Quadrature observables Unsharp quadrature observables Phase space observables Radon transform Tomography observable Statistical method of moments 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Accardi, L.: Some trends and problems in quantum probability. In: Accardi, L., Frigerio, A., Giorni, V. (eds.) Quantum Probability and Applications to the Quantum Theory of Irreversible Processes. Lecture Notes in Mathematics, vol. 1055, pp. 1–19. Springer, Berlin (1984) CrossRefGoogle Scholar
  2. 2.
    Albini, P., De Vito, E., Toigo, A.: Quantum homodyne tomography as an informationally complete positive-operator-valued measure. J. Phys. A, Math. Theor. 42, 295302 (2009) CrossRefGoogle Scholar
  3. 3.
    Ali, S.T., Doebner, H.D.: On the equivalence of nonrelativistic quantum mechanics based upon sharp and fuzzy measurements. J. Math. Phys. 17, 1105–1111 (1976) CrossRefMathSciNetADSGoogle Scholar
  4. 4.
    Ali, S.T., Prugovečki, E.: Classical and quantum statistical mechanics in a common Liouville space. Physica A 89, 501–521 (1977) CrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Bohr, N.: The quantum postulate and the recent development of atomic theory. Nature 121, 580–590 (1928) MATHCrossRefADSGoogle Scholar
  6. 6.
    Bohr, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 48, 696–702 (1935) MATHCrossRefADSGoogle Scholar
  7. 7.
    Busch, P., Cassinelli, G., Lahti, P.: On the quantum theory of sequential measurements. Found. Phys. 20, 757–778 (1990) CrossRefMathSciNetADSGoogle Scholar
  8. 8.
    Busch, P., Grabowski, M., Lahti, P.J.: Operational Quantum Physics. Springer, Berlin (1997). 2nd corrected printing Google Scholar
  9. 9.
    Busch, P., Kiukas, J., Lahti, P.: Measuring position and momentum together. Phys. Lett. A 372, 4379–4380 (2008) CrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Busch, P., Lahti, P.: The complementarity of quantum observables: theory and experiments. Riv. Nuovo Cimento 18(4), 1 (1995) CrossRefMathSciNetGoogle Scholar
  11. 11.
    Busch, P., Lahti, P., Mittelstaedt, P.: The Quantum Theory of Measurements. Springer, Berlin (1996). 2nd revised edn. Google Scholar
  12. 12.
    Carmeli, C., Heinonen, T., Toigo, A.: On the coexistence of position and momentum observables. J. Phys. A, Math. Gen. 38, 5253–5266 (2005) MATHCrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Cassinelli, G., D’Ariano, G.M., De Vito, E., Levrero, A.: Group theoretical quantum tomography. J. Math. Phys. 41, 7940–7951 (2000) MATHCrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Cassinelli, G., Varadarajan, V.S.: On Accardi’s notion of complementary observables. Infinit. Dimens. Anal. Quantum Probab. Relat. Top. 5, 135–144 (2002) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Davies, E.B., Lewis, J.T.: An operational approach to quantum probability. Commun. Math. Phys. 17, 239–260 (1970) MATHCrossRefMathSciNetADSGoogle Scholar
  16. 16.
    Dvurečenskij, A., Pulmannová, S.: Uncertainty principle and joint distributions of observables. Ann. Inst. Henri Poincaré Phys. Thoer. 42, 253–65 (1985) MATHGoogle Scholar
  17. 17.
    Kiukas, J., Lahti, P.: A note on the measurement of phase space observables with an eight-port homodyne detector. J. Mod. Opt. 55, 1891-1898 (2008) MathSciNetADSGoogle Scholar
  18. 18.
    Kiukas, J., Lahti, P., Pellonpää, J.-P.: A proof for the informational completeness of the rotated quadrature observables. J. Phys. A, Math. Theor. 41, 175206 (2008). (11pp) CrossRefADSGoogle Scholar
  19. 19.
    Kiukas, J., Lahti, P., Schultz, J.: Position and momentum tomography. Phys. Rev. A 79, 052119 (2009) CrossRefADSGoogle Scholar
  20. 20.
    Kraus, K.: Complementary observables and uncertainty relations. Phys. Rev. D 35, 3070–3075 (1987) CrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Lahti, P., Pellonpää, J.-P.: Covariant phase observables in quantum mechanics. J. Math. Phys. 40, 4688–4698 (1999) CrossRefMathSciNetADSGoogle Scholar
  22. 22.
    Leonhardt, U.: Measuring the Quantum State of Light. Cambridge University Press, Cambridge (1997) Google Scholar
  23. 23.
    Ludwig, G.: Foundations of Quantum Mechanics I. Springer, Berlin (1983) MATHGoogle Scholar
  24. 24.
    Pellonpää, J.-P.: Quantum tomography, phase space observables, and generalized Markov kernels. J. Phys. A, Math. Theor. 42, 465303 (2009) CrossRefADSGoogle Scholar
  25. 25.
    Reichenbach, H.: Philosophic Foundations of Quantum Mechanics. University of California Press, Berkeley (1944) Google Scholar
  26. 26.
    von Neumann, J.: Die Eindeutigkeit der Schrödingerschen Operatoren. Math. Ann. 104, 570–578 (1931) CrossRefMathSciNetGoogle Scholar
  27. 27.
    von Weizsäcker, C.F.: Quantum theory and space-time. In: Lahti, P., Mittelstaedt, P. (eds.) Symposium on the Foundations of Modern Physics, pp. 223–237. World Scientific, Singapore (1985) Google Scholar
  28. 28.
    Ylinen, K.: On a theorem of Gudder on joint distributions of observables. In: Lahti, P., Mittelstaedt, P. (eds.) Symposium on the Foundations of Modern Physics, pp. 691–694. World Scientific, Singapore (1985) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Turku Centre for Quantum Physics, Department of Physics and AstronomyUniversity of TurkuTurkuFinland

Personalised recommendations